Recently in Calculus Category

Integration, Quasi Mode

Last time we saw how it was possible to use uniformly distributed random variables to approximate the integrals of univariate and multivariate functions, being those that take numbers and vectors as arguments respectively. Specifically, since the integral of a univariate function is equal to the net area under its graph within the interval of integration it must equal its average height multiplied by the length of that interval and, by definition, the expected value of that function for a uniformly distributed random variable on that interval is the average height and can be approximated by the average of a large number of samples of it. This is trivially generalised to multivariate functions with multidimensional volumes instead of areas and lengths.
We have also seen how quasi random sequences fill areas more evenly than pseudo random sequences and so you might be asking yourself whether we could do better by using the former rather than the latter to approximate integrals.

Clever you!

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Monte Carlo Or Bust

We have taken a look at a few different ways to numerically approximate the integrals of functions now, all of which have involved exactly integrating simple approximations of those functions over numerous small intervals. Whilst this is an appealing strategy, as is so often the case with numerical computing it is not the only one.

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The Nesting Instinct

We have spent some time now looking at how we might numerically approximate the integrals of functions but have thus far completely ignored the problem of integrating functions with more than one argument, known as multivariate functions.
When solving integrals of multivariate functions mathematically we typically integrate over each argument in turn, treating the others as constants as we do so. At each step we remove one argument from the problem and so must eventually run out of them, at which point we will have solved it.
It is consequently extremely tempting to approximate multivariate integrals by recursively applying univariate numerical integration algorithms to each argument in turn.

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A Jump To The Left And Some Steps To The Right

We have previously seen how we can approximate the integrals of functions by noting that they are essentially the areas under their graphs and so by drawing simple shapes upon those graphs and adding up their areas we can numerically calculate the integrals of functions that we might struggle to solve mathematically.
Specifically, if we join two points x1 and x2 on the graph of a function f with a straight line and another two vertically from them to the x axis then we've drawn a trapezium.

In regions where a function is rapidly changing we need a lot of trapeziums to accurately approximate its integral. If we restrict ourselves to trapeziums of equal width, as we have done so far, this means that we might spend far too much effort putting trapeziums in regions where a function changes slowly if it also has regions where it changes quickly.

The obvious solution to this is, of course, to use trapeziums of different widths.

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The Tip Of The Romberg

Last time we took a first look at the numerical approximation of integrals with the trapezium rule, a rather simplistic algorithm with the unfortunate property that its accuracy was implicitly, rather than explicitly, specified.
As a user of numerical libraries, including my own, I would much rather have an algorithm that works to a given accuracy, or at the very least tries to, than have to figure it out for myself and I suggested that we might apply the lessons learned from our attempts to numerically approximate derivatives to do just that.

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Trapezium Artistry

We have covered in some detail the numerical approximation of differentiation but have yet to consider the inverse operation of integration and I think that it's high time that we got around to it...

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Like Robinson's non-standard numbers, Conway's surreal numbers extend the reals in a way that admits a rigorous definition of infinitesimals.
So what's to stop us from actually using infinitesimals in our calculations?
Nothing. That's what!

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One of the surprising facts about differentiation is that it is almost always possible to find the expression for the derivative of a function if you have the expression for the function itself. This might not seem surprising until you consider the inverse operation; there are countless examples where having the expression for the derivative doesn't mean that we can find the expression for the function.
This is enormously suggestive of a method by which we can further improve our calculation of derivatives; get the computer to generate the correct expression for us.

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We began this arc of posts with a potted history of the differential calculus; from its origin in the infinitesimals of the 17th century, through its formalisation with Analysis in the 19th and the eventual bringing of rigour to the infinitesimals in the 20th.
We then covered Taylor’s theorem and went on to use it in a comprehensive analysis of finite difference approximations to the derivative in which we discovered that their accuracy is a balance between approximation error and cancellation error, that it always depends upon the unknown behaviour of higher derivatives of the function and that improving accuracy by increasing the number of terms in the approximation is a rather tedious exercise.
Of these issues, the last rather stands out; from tedious to automated is often but a simple matter of programming. Of course we shall first have to figure out an algorithm, but fortunately we shall be able to do so with relative ease using, you guessed it, Taylor’s theorem.

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In the previous post I discussed the foundations of the differential calculus. Initially defined in the 17th century in terms of the rather vaguely defined infinitesimals, it was not until the 19th century that Cauchy gave it a rigorous definition with his formalisation of the concept of a limit. Fortunately for us, the infinitesimals were given a solid footing in the 20th century with Conway’s surreal numbers and Robinson’s non-standard numbers, saving us from the annoyingly complex reasoning that Cauchy’s approach requires.

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